If a conductor - a long rod - moves at constant speed across the "lines" of a uniform magnetic field, is there an EMF within this conductor? Or, if a conducting rod rotates at uniform rate, pivoted in the middle or at one of its ends in a uniform magnetic field perpendicular to the plane of rotation, is there an EMF generated within the conductor?
If the setup you have in mind is like the image below, there will be a voltage across the conductor.
This is due to the fact that there no closed path for a current.
The mobile electrons in the conductor "feel" a magnetic force towards the b end of the rod.
The electrons "bunch" up at that end resulting in an electric field that points from a to b.
Assuming the velocity is constant, the force on the electrons due to the electric field cancels the magnetic force.
An EMF is generated whenever a conductor moves relative to a magnetic field so that the conductor is cutting across the magnetic field lines. The EMF generated is the cross product of the magnetic field and the motion of the conductor (I may have the sign flipped, don't remember off the top of my head).
To answer your rotating rod question: yes EMF is produced inside the rod. However, note that this EMF will not be end to end of the rod, but from each end to its center.
If you attached brushes to the ends of the rod such that they touched the inside of a fixed cylinder, then the whole apparatus would be a generator between this cylinder and the center of the rod. If you connected this generator to a load, current would now flow, which now produces a force on the rod, which would oppose its motion.
Faraday's law relates to the amount of energy a charge would gain by going around a loop. This energy per unit charge is called an "EMF". Because there is no change in the flux through any surface in this situation, there is no loop that gives a charge a different energy.
This doesn't mean that there is no path that gives the particle a different energy. As Alfred Centauri's answer illustrates, a conductor moving through a magnetic field is subject to the Hall effect, which will create a voltage between the two ends. This is not a violation of faraday's law, because the closed-loop voltage remains zero.